Abstract
We provide a brief and elementary review on the mathematical theory of stochastic processes. In particular, the correspondence between stochastic differential equations (SDEs) and master equations (MEs) is studied in detail. We first consider the Liouville equation as the simplest ME for an ordinary differential equation, and we next consider the MEs for SDEs driven by various Poisson noises. The Gaussian noise is introduced as a limit from the symmetric Poisson noise, and the Fokker-Planck equation is derived for an SDE driven by the Gaussian noise. We finally address an SDE driven by state-dependent Gaussian and Poisson noises, and studies the most general MEs in Markovian stochastic processes.
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